Problem: Let $z$ be a complex number such that $|z| = 2.$  Find the largest possible distance between $(3 + 4i)z^3$ and $z^5$ when plotted in the complex plane.
We want to maximize
\[|(3 + 4i)z^3 - z^5| = |z^3| |3 + 4i - z^2| = |z|^3 |3 + 4i - z^2| = 8 |3 + 4i - z^2|.\]In other words, we want to maximize the distance between $3 + 4i$ and $z^2.$

Since $|z| = 2,$ the set of complex numbers of the form $z^2$ lie on a circle with radius $|z|^2 = 4.$  The distance between $3 + 4i$ and $z^2$ is maximized when $z^2$ lies on the line passing through the origin and the $3 + 4i.$  (This line intersects the circle at two points, so we take the one that is farther from $3 + 4i.$)

[asy]
unitsize(0.5 cm);

draw(Circle((0,0),4));
draw((-4.5,0)--(4.5,0));
draw((0,-4.5)--(0,4.5));
draw((0,0)--(3,4));
draw((0,0)--(-4/5)*(3,4));

label("$4$", (-4/5)*(3,4)/2, NW);

dot("$3 + 4i$", (3,4), NE);
dot("$z^2$", (-4/5)*(3,4), SW);
[/asy]

For this number, the distance between $3 + 4i$ and $z^2$ is $4 + 5 = 9,$ so the maximum value of $8 |3 + 4i - z^2|$ is $8 \cdot 9 = \boxed{72}.$